Inverse Problems 6 (1990) 247-266. Printed in the UK
The role of the Born approximation in nonlinear inversion
Roe1 Snieder
Department of Theoretical Geophysics. University of Utrecht, PO Box 80.021, 3508 TA
Utrecht. The Netherlands
Received 12 May 1989, in final form 7 November 1989
Abstract. A perturbation analysis of nonlinear inversion is presented. As a prototype of
nonlinear inversion, inverse scattering with the Marchenko equation is considered from a
perturbative point of view. It is shown that inverse scattering methods using the Marchenko
equation implicitly reconstruct the potential by removing the nonlinearities from the data,
and by performing a Born inversion of the resulting linear component in the data (the first
Born approximation). This is illustrated with a one-dimensional example. This interpretation
of the mechanism of inverse scattering algorithms clarifies the ‘miracle of Newton’, and has
profound consequences for both the theoretical and the practical aspects of inverse scattering, in particular for the stability of inverse scattering schemes. As shown in several examples,
the arguments presented for inverse scattering with the Marchenko equation can be
generalised to a wide class of nonlinear inverse problems.
1. Introduction
Inverse scattering methods such as the Marchenko or the Gel’fand-Levitan algorithms
belong to the rare examples of an exact formulation of a nonlinear inverse problem.
Despite the simple mathematical formulation of these algorithms, surprisingly little is
known about the mechanism by which these algorithms actually perform the inversion.
Which characteristics of the data are actually used for the reconstruction of the unknown
potential? How are the nonlinearities in the problem handled? This paper clarifies these
issues by showing a perturbative analysis of exact inverse scattering methods.
This pertubative analysis is presented for the one-dimensional (ID) Marchenko
equation, which forms the solution for the inverse problem for the I D plasma wave
equation (Balanis 1972, Burridge 1980). However, the obtained results depend only on
the general structure of the Marchenko equation, but not on its details. This means that
the results of this paper can be generalised to any nonlinear inversion scheme for which
the forward problem has a regular perturbation expansion, and for which the inverse
problem can be formulated in terms of operators that act repeatedly on the data. The
main conclusion of the perturbative analysis is that only the linear components of the
data (the first Born approximation, or the single reflected waves) contribute to the
reconstruction of the potential. Implicitly the nonlinear components are subtracted from
the data by solving the integral equation of inverse scattering.
This observation gives new insights in nonlinear inversion algorithms. It makes it
easy to understand why inverse scattering methods suffer from stability problems
(Koehler and Taner 1977), why exact inverse scattering methods for media with a
variable wave velocity have only been formulated relative to a fast reference medium
0266-5611/90/020247 + 20 $3.50
1990 IOP Publishing Ltd
247
248
R Snieder
(Cheney et a1 1989), and it gives an example of the remarkable ‘miracle of Newton’ for
inverse scattering in three dimensions.
The I D plasma wave equation and the Marchenko equation are presented in section 2.
In the subsequent section a perturbative analysis is applied to the Marchenko equation.
This is illustrated in section 4 with a numerical example. Section 5 features three
examples of a perturbative analysis of other nonlinear inversion schemes (the ‘miracle’
of Newton, the inversion of Corones et al(l983) using invariant embedding, and the
iterative inversion of Morawetz and Kriegsmann (1983)). In an appendix it is shown
explicitly that the quadratic nonlinearities for the 3~ Schrodinger equation are correctly
subtracted from the scattering data by the Newton-Marchenko algorithm.
2. Inverse scattering for the
1~
plasma wave equation
The perturbative analysis of inverse scattering algorithms is the same for all inverse
scattering schemes which involve the solution of an integral equation similar to the
Marchenko equation or the Gel’fand-Levitan equation. Rather than giving a general
derivation, the perturbation analysis is presented for the inversion of the I D plasma wave
equation (PWE) using the Marchenko equation. The arguments presented in this paper
are easily generalised for other inverse scattering equations such as presented by Rose
et a1 (1986).
The problem is considered where the wavefield u(x,t ) satisfies the PWE:
uxr(x, 1)
-
urr(x,t )
-
V(X>U(X,
t) = 0
(1)
where V(x) is an unknown potential. For simplicity it is assumed that
V(x) = 0
for x < 0.
Furthermore it is assumed that V e L 2 and that (Chadan and Sabatier 1989):
(2)
This potential is being probed by an impulsive wave coming in from the left, so that:
u(x,t ) = s(t - x)
for t < 0.
(4)
The waves reflected by the potential are recorded at x = 0, so that the data for the
inversion are given by the reflection time series R(t):
R(t) = U(X = 0, t ) - d(t).
(5)
The inverse problem consists of the determination of the unknown potential V ( x )given
the reflections R(t).
The solution of this inverse problem has been formulated both in the spectral domain
(Agranovich and Marchenko 1963) and in the time domain (Balanis 1972, Burridge
1980). The reflection time series R(t) serves as an integral kernel in the Marchenko
equation:
K(x,t )
+ R(x + t ) +
irr
K(x,z)R(z + t)dz
= 0.
(64
In shorthand notation this equation is also written as
K + R + K R = 0.
(6b)
The Born upproximution in nonlinear inversion
249
Once K ( x , t ) has been determined from the Marchenko equation, one can find the
potential using
V ( x )= 2
dK(x, x)
dx
(7)
____
’
3. A perturbative analysis for the forward and inverse problem of the
wave equation
ID
plasma
In order to understand the role of the nonlinearities in inverse scattering it is necessary
to distinguish the linear effects from the nonlinear effects. For this reason it is convenient
to attach a coupling parameter E to the potential:
(8)
V = EV(X).
A perturbation expansion of the forward problem can be obtained by expressing the
solution of (1) with the incident wave (3) as the following integral equation:
U(X, t) = d ( t - X)
+E
c c
dx’
dt‘ G,(x, t; x’, ~ ’ ) V ( X ’ ) U (t’)
X’,
with Go the causal Green function of the unperturbed
d,,G,(x, t;x‘, t’)
-
(9)
PWE:
~ , , G , ( Xt;x‘,
, t’) = 6(x - x’)6(t - t’).
(10)
This Green function is given by
G,(x, t;x’, t’) = -+H(t - t‘ - Ix - x’l)
(1 1)
H(t) being the Heaviside function.
Iterating the integral equation (9) leads to a Neumann series which can be considered
as a Taylor expansion in E . Considering the resulting Neumann series for x = 0 leads
with equation ( 5 ) to a perturbation expansion for the reflected waves:
+
+
R(t) = &R,(t) &*R2(t) . . .
(12)
the R, being given by:
dx, dt, Go(x = 0, t ; x,, t , ) V ( x ,) d ( t , - x I )
R,(t) = ~ d x l d t , d x , d t , d x 3 d t , G o (=
x O,t;xl,t,)V(x,)
(13c)
with obvious higher-order generalisations. The term R I(t) is the part of the data which
depends linearly on the potential (the single reflected waves). In analogy with the
quantum mechanical nomenclature this term will be referred to as the Born approximation. R2(t) describes the wave that experienced two interactions with the potential,
etc.
250
R Snieder
Just as the reflection R(t),the solution K(x, t ) of the Marchenko equation (5) is also
a nonlinear function of the potential; this function can also be expanded in a Taylor
series in E :
+
+
K(x,t) = ~ K , ( x , t ) E * K ~ ( x , ~. ). . .
(14)
Inserting the Taylor expansions (12) and (14) in the Marchenko equation (6), and
equating the coefficients of equal powers of E one obtains a hierarchy of equations for
K,, :
K1 -- - R ,
K2
=
-R2
KS
=
-
(154
R,
-
KIR,
(15b)
-
K , R, - K2Rl
(15c)
These equations constitute the relation between the K, and the terms R, in the Neumann
series. A recursive application of these relations allows for the elimination of the Kn from
the right-hand side of (15d). This gives
KI = - RI
( 16 4
+ RI R ,
K, = - R3 + RZR, + RlR2K2
= - R,
n
K,, =
2
in= I
(166)
R,R,R,
(16c)
RI, . . .
(-1>”’
I,+
+i,,=n
For example, K2(x, x ) is given explicitly by
K2(x,x)=
-
R2(2x)+
1‘
R:(x + r)dr.
--I
Alternatively, one can insert (8) and the Taylor expansion (14) in the relation (7)
between V and K. Equating the coefficients of equal powers of E gives
dKl (x,x)= V ( x )
2 __
dx
dKn ( x , x ) = 0
2dx
(1 7 4
n 2 2.
(17b)
The above expressions imply that V ( x )is completely determined by K , . According to
(15a), the first-order contribution K , ( x ,x) depends only on the first Born approximation
R I ,and is thus independent of the nonlinear components in the data. This means that
only the first Born approximation contributes to the reconstruction of the potential, and
that the nonlinear components in the data do not contribute to the reconstruction of the
potential.
So what happens to the nonlinear components R, (n 2 2) in the data? As an example,
consider the quadratic term K2(x,x)in equation (1%). This term consists of the difference between the second Born term R,, and the iterated term K , R , . According to
(17b), K2 does not give a net contribution to the reconstruction of the potential, which
The Born approximation in nonlinear inversion
25 1
means that the second Born approximation R, is being subtracted from the data by the
iterated term K , R , . It can be seen explicitly in (16e) that the second-order Born term R2
is being cancelled by an iteration of the first Born approximation R, . The same mechanism is operative for the higher-order terms. For example, the third Born approximation R, is being cancelled in (16c) by repeated iterations of the lower-order Born
approximations R , and R,.
This means that in the inversion using the Marchenko equation, only the first Born
approximation contributes to the reconstruction of the potential, and that in the inversion the nonlinearities are being subtracted from the data. Note that this does not imply
that one need only perform a Born inversion of the data in order to obtain the potential.
The reason for this is that one does not measure the first Born approximation; the data
consist of a sum of both the linear components and the nonlinear components. One really
needs to go through a nonlinear inversion, such as the Marchenko method, in order to
eliminate the effect of the nonlinearities.
One can show from the integral representation for K(x,t)(Burridge 1980) that
K(0,O) = 0. With (17a, b ) this implies that, for x > 0,
n 3 2.
K,(X,x) = 0
(1 8)
4. A numerical example
In this section a numerical example is presented to show that the nonlinear components
are correctly subtracted from the data by using the Marchenko equation. In these
examples the potential shown in figure 1 is used. The precise form of the potential is not
important, but it should be kept in mind that the potential consists essentially of two
scattering zones, and that waves with wavelengths much shorter than the distance
between the scattering zones can bounce back and forth between the two sides of the
potential. The low-frequency components in the wavefield interact in a simultaneous way
with both sides of the potential.
The first three terms (1 3a-c) of the Neumann series are shown together with their sum
in figure 2 . The Born approximation R , consists of reflections with opposite polarity
from the two sides of the potential. The third-order contribution R3 shows a distinct
I
0.004 -
.-'p
+
0.-
+
U
I
a"
-0.004
0
\
50
I ,
100
Oist m e
150
2oo
Figure 1. Potential used in the numerical
example.
R Snieder
252
0.02
VI
.-L
VI
E
-0.02
0
100
200
300
Time
Figure 2. The first (thin full line), second
(broken line) and third (dotted line) Born
approximations and their sum (thick full
line).
wave arrival around t = 280, this is the wave that has bounced back and forth once
between the sides of the potential. The static offset of the signals reflects the fact that no
fixed boundary values are imposed on u(x,t ) ; this behaviour is the same as for an infinite
stretched string without fixed boundary conditions.
The first-order contribution K,(x,x)
computed with (150) and (13a) is shown in
figure 3. Application of the differentiation (17a) of this function leads to a result which
is indistinguishable from the original potential of figure 1. This reflects the fact that it
is only the first Born approximation which gives a non-zero contribution to the
reconstruction of the potential.
The second-order contributions to K, (x,x) computed with (1 3a,b) and (1 5b) are
shown in figure 4. The second Born approximation R, is exactly cancelled by the repeated
iteration K , R I(= - R , R , ) of the first Born approximation, which verifies that the
quadratic component in the data is correctly subtracted. Note that this does not mean
that K2(x,t ) vanishes for all t # x, since the condition (18) only implies that K2(.x,x)
vanishes. However, the latter condition is indeed satisfied.
Figure 5 features the cubic contributions to K3(x,x)
computed with (13a-c)
and (15c). Again it can be seen that the third-order Born approximation R, is being
cancelled by repeated iterations of lower-order Born approximations. The total
x) is not quite zero; the slight deviation around x = 100 is due to
contribution to K3(x,
numerical errors. Note that the wave which has bounced back and forth between the
0.02 -
2-
0 -
I
k-
-0.02 -
0
50
100
Distance
150
Figure 3. The first-order contribution
K, (x, x) to the reconstruction of the
potential. The derivative 2dK, ( x , n)/d.x is
indistinguishable from the true potential in
figure 1.
The Born approximution in nonlinear inversion
0.02
-0.02
i
1
I
0
50
100
2
150
Distance
0.02
253
Figure 4. The second-order contributions
- R2 (chain line) and - K , R , (broken line)
to the reconstruction of the potential, with
their sum K 2 ( x , x )(full line).
i
-0.02
0
50
100
2 1
150
Distance
Figure 5. The third-order contributions
- R, (chain line), - K , R , (broken line)
and - K , R , (dotted line) to the reconstruction of the potential. with their sum
K 3 ( x ,s) (full line).
sides of the potential (the valley around x = 150) is eliminated so that it does not
contribute to the reconstruction of the potential.
In practical inversions one does not, of course, known which parts of the data are due
to linear effects, and which parts are caused by nonlinearities. Therefore one should
really solve the Marchenko equation, or use some other nonlinear inversion scheme. As
noted by Ge (1987), the Marchenko equation can efficiently be solved by iteration. In this
algorithm the starting value is
R"'(x,t ) = - R(x,t )
(1 9 4
and the following iterations are defined by
l?("+')(x,t ) =
- R ( x , t) -
' j?(nl(x,.r)R(.r+ t)dz.
-I
(1 96)
If this process converges, the final solution satisfies the Marchenko equation (6a). It
should be noted that I?) and K, are fundamentally different functions, K,, depends by
definition (14) only on the potential in the nth power, whereas Z? contains scattering
effects of different orders.
The algorithm of Ge (1987) applied to the scattering data for the I D potential of
figure 1 is shown in figure 6. In the iterations shown only the terms up to third order are
R Snieder
254
50
0
100
150
i
Distance
lo
Figure 6. Iterative solution of the
Marchenko equation after one (dotted
line), two (broken line). and three iterations
(full line). Only terms up to third order
have been taken into account.
included. (For simplicity, K ( x , x ) is shown rather than its derivative.) After the first
iteration, the left bump of the potential (around x = 30) is already reconstructed quite
well, but the second bump (around x = 80) and the area to the right are poorly
reconstructed. The reason for this is that the first iteration of the Marchenko equation
is equivalent to doing a Born inversion of the full data including the nonlinearities. The
effect of the nonlinearities becomes stronger in the later parts of the signal (because then
the waves have had time to bounce back and forth), so that after the first iteration the
reconstruction suffers most from incorrectly treated nonlinearities further to the right.
However, in subsequent iterations these incorrectly handled nonlinearities are subtracted
from the reconstruction. The reconstruction after three iterations gives, after differentiation, exactly the potential. In this simple example, three iterations were sufficient to
reconstruct the true potential because only terms up to third order were taken into
account. However, when higher-order nonlinearities are present in the data one has to
perform more iterations.
It is possible to show that after M iterations all the multiple scattering effects up to
nth order are correctly removed from the reconstructed potential, and that the error is
of order E " + ' . In order to see this, insert the Neumann series (12) in the algorithm (19a, 6);
this gives, in the abbreviated notation of section 3 ,
k"= -&RI
-
&*R,- E ~ R+, O(e4)
gc2)
= --ER,- c2(R,- R , R , ) - E,(R,- R IR, - R,R,) + O ( E ~ )
=
-&RI -- &,(R,- R , R , ) - e3(Rj - R IR2 - R,R,
+ R IR IR , ) + O ( E ~ ) .
(2Oa)
(20c)
Using equation (1 6) one finds that
E'')= EK,+ O(E')
(214
k2'= E K ,+ c2K2+ O(E3)
k(j)= BK, + E'K, + e3K3+ O ( E ~ )
( 2 16)
( 2 1c)
with obvious generalisations to higher order. Applying the differentiations (1 7a) and
(1 7b) one finds that
di?'")(x, x)
= &V(X) O(&fl+l).
dx
+
255
The Born approximation in nonlinear inversion
This means that after n iterations, all nonlinear scattering effects up to nth order are
correctly subtracted from the data and d o not contribute anymore to the reconstructed
potential. The number of iterations needed in practice depends on the degree of nonlinearity and the required accuracy. For example, with seismic reflection data one can
sometimes clearly identify a set of multiply reflected waves. If the Nth multiple is of the
same order as the noise level, one needs to perform N iterations.
5. Generalisations to other nonlinear inversion methods
The preceding theory was developed for the I D plasma wave equation. However, the
arguments employed for the perturbation analysis of the forward and the inverse
problem did not rely in an essential way on the details of the PWE or the Marchenko
equation. The only essential ingredient in the analysis was that the forward problem
could be expressed as a regular series expansion in E , and that the inversion could be
expressed in the form of operators that act repeatedly on the data. A formal proof of this
statement can be found in Snieder (1990), where a perturbation analysis is applied to
nonlinear inversion in general. Rather than repeating that proof, three examples are
presented in the following subsections of well known nonlinear inverse scattering
problems for which only the first Born approximation contributes to the reconstruction
of the inhomogeneity, and where the nonlinearities are removed in the inversion.
5.1. Example 1. Inverse scattering,for the 3D Schrodinger equation and the ‘miracle
of Newton’
An inverse scattering algorithm for the three-dimensional Schrodinger equation has been
formulated by Newton (1980, 1981). In this method, one needs to solve an integral
equation similar to the Marchenko equation (6). Just as in the one-dimensional case, the
integral kernel for this integral equation depends on the scattering data. The potential
follows by a differentiation of the solution of the integral equation. Therefore, the
arguments of the preceding sections apply equally well to the Newton-Marchenko
method.
The Schrodinger equation in three dimensions is given by
(V2 + k‘)$(k, it, Y) - E V(r)$(k, it, Y)
= 0.
(23)
It is assumed that V ELz and that positive numbers C and a exist, such that
lV(v)l d C ( a
+ IWP
lVV(Y)l
< C(a + Ivl)-’
(24)
with ,U > 3 and 1’ > 7/2 (Chadan and Sabatier 1989). If the potential is irradiated with
an incident plane wave, the wavefunction in the far field is asymptotically given by:
$(k, A, Y) = exp(ikit * v)
+ A,(?, A) exp(ikr)
r
(25)
~
A , being the scattering amplitude. The unit vector A denotes the direction of the incoming
plane wave with wavenumber k.
The Newton-Marchenko method proceeds by computing for every point Y, for every
incident A and every final it’, an integral kernel R from the scattering data A,:
;
1
r x
R(E,it, it’, v ) = 24n2
--J
dk kA,( - it’, ii) exp( - ik[r
+ (it + it’)
* Y]}.
(26)
256
R Sniedev
This integral kernel is then used in the Newton-Marchenko equation
c
d2n’R(r,ii, ii’, v )
+
c 7
c
d2n’R(x
+ j,ii, ii’, v)K(B,ii‘, r ) .
(27)
In this section and the appendix, Jd’n’ denotes an integration over the unit sphere. The
potential is finally recovered by differentiation:
V(v) = - 2;
*
VK(x = O,ii,V).
(28)
There are two aspects of the Newton-Marchenko equation which are not well
understood. First, the right-hand side of equation (28) depends on the unit vector ii,
whereas the left-hand side is independent of ii. This is called the ‘miracle of Newton.’
Equation (28) should be true if A k is indeed caused by scattering by a local potential.
However, for general functions A , , the miracle need not necessarily be satisfied (i.e. the
right-hand side of (28) depends on ii and in this case no local potential exists but a
non-local potential exists when equation (28) still has a solution).
Second, there is the paradox that the Newton-Marchenko equation requires scattering data for all directions of incidence, for all scattering directions, and for all energies.
However, for a Born inversion it suffices to measure a more restricted set of scattering
data. (For example, it can be inferred from (30a) that a Born inversion of the potential
can be performed by using the scattering amplitude for only one angle of incidence. all
energies and all scattering directions.) It therefore appears that the Newton-Marchenko
method requires a redundant data set. Note that this redundancy argument applies to
the Born inversion, but not necessarily to the nonlinear inversion. The point of view of
nonlinear inversion presented in this paper sheds light on both paradoxes. It is therefore
instructive to d o a perturbative analysis of the Newton-Marchenko method. This
analysis is analogous to the analysis presented in section 3 for the one-dimensional PWE.
The scattering amplitude can be expanded in the following Born series (Rodberg and
Thaler 1967):
with
J
Ai’)(ii’,ii) = d’r exp( - ikii’ * v ) V(v)exp(ikii
-4n
-I
[ { d3 exp(
167~’
Af) (i
f i )’
=
,
- d3v
Y’
-
ikii’ v ) V(v)
(300)
U)
exp(ik1r - r‘ 1)
/v - v l I
-
V(v’)exp(ikii v ’ ) .
(30b)
The fact that it is assumed that the scattering amplitude can be written as the Born series
for a local potential, implies that the analysis presented in this section has no bearing on
the characterisation problem; it only has implications for the construction problem. The
solution K from the Newton-Marchenko equation (27) is implicitly a function of the
potential and can be expanded in a Taylor series as in equation (14). The first-order
contribution K , follows by applying the equivalent of (1 5a) to the Newton-Marchenko
equation (27):
K , (a, A,x)
=
d2n’R,( x , ii, ii’, x),
257
The Born upproxinzution in nonlinear inversion
where R , is obtained by inserting the first Born approximation (300) in (26). This gives,
after carrying out the ii' integration,
Putting x
K , (X
=
0 and performing the k integral leads to
= O,ii,x) =
XI
xi
-ii.(u-
+ ii * ( r -
(33)
Replacing the integration variable r by r x, switching to polar coordinates and
carrying out the integration over the angles gives
+
1
K,(R =
f X
4J"
O,ii,x) = -
dr(V(x + rii) - V ( x - rii))
(34)
+ rii)
(35)
so that
- 2A V, K , (R = 0, ii, x) =
Px'
-
+ii
*
J
0
dr (V, V ( x
-
V, V ( x - rii)).
In this expression V, denotes the gradient with respect to the x coordinates. Introducing
the vector y = rii, and converting the x differentiations to y differentiations gives
- 2 i i .V,K,(R= O,ii,x) =
iJdy . (V, V ( X+ y ) + V, V ( X- y ) ) .
1
-
friX
0
(36)
Since the potential is assumed to vanish at infinity, this gives
- 2ii
v, K , (.
0, 3, x)
V(x)
(37)
which confirms the occurrence of the miracle (28). The right-hand side of (36) is indeed
independent of ii, the reason being that the potential only gives a contribution in the
pointy = 0 (which is independent of ii) and in the points y = fiix (where the potential
vanishes). In the derivation we can thus see the miracle 'in action' in a situation where
the miracle (28) should hold.
In an analogous way to (17b), the higher-order terms K, (n 2 2) satisfy
- 2ii * V, K,, ( x = 0, ii, x) = 0, so that these terms do not contribute to the reconstruction
of the potential. This relation is explicitly verified for the quadratic terms in the appendix. The first iteration of the first Born approximation is thus the only term which
contributes to the reconstruction of the potential; this means that the mechanism by
which the miracle occurs in the construction of the potential is explained by the above
derivation. However, note that it is assumed in (30) that the data A , are the Born series
of some local potential. If this is not the case the argument of this section breaks down,
and the miracle need not occur. The treatment of this section does not address the
characterisation problem, which is a critical issue for the occurrence of the miracle.
The problem of the apparent redundancy of the Newton-Marchenko method can
also be understood with the results of this paper. It is true that the Newton-Marchenko
method requires more data than are needed for a Born inversion. However, according
to the results of section 3, two steps are implicitly taken in nonlinear inversion. The
nonlinearities are removed from the data, and a Born inversion of the linear components
of the data is performed. It is conceivable that the removal of the nonlinearities requires
more data than are needed for a Born inversion of the linear component in the data. The
*
=
=
258
R Snieder
argument that the Newton-Marchenko algorithm requires more data than a Born
inversion does not necessarily imply that it requires a redundant data set, because it is
possible that the removal of the nonlinearities requires a larger data set than is needed
for a Born reconstruction of the potential.
5.2. Example 2. Nonlinear inversion using invariant embedding
An alternative to the inverse scattering methods using integral equations is formulated
using Ricatti equations (Gjevik et a1 1976, Corones et a1 1983, Bregman et a1 1985).
Corones et a1 (1983) consider reflected waves recorded at x = 0 for the following I D
equation:
+ &V(X)U,= 0 .
(38)
U,, - U,,
It is assumed that the potential extends from x = 0 to x = X . They analyse this equation
using an invariant embedding approach where ones studies a suite of truncated potentials extending from an arbitrary point x to the right side X of the true potential. The
reflected waves R(x, t ) then depends on the position x of the left point of truncation. For
details the reader is referred to Corones et a1 (1983) who show that R(x, t ) satisfies the
following Ricatti equation
a,R(x, t ) - 2d,R(x, t ) =
-
+ & V ( XJ) R(x,s)R(x, t
0
-
s)ds
(39)
with boundary conditions
R(x,0) = -;e V ( x )
(40a)
R ( X , t ) = 0.
(40b)
For the forward problem one specifies V ( x ) ,and hence by virtue of (40a), R(x,0),
and one integrates (39) leftwards towards x = 0 to obtain the reflected waves for the
untruncated medium:
r ( t ) = R(0,t ) .
(41)
In the inverse problem one specifies r(t), and hence R(O,t), and one integrates (39)
rightwards towards x = X . The potential follows then from (40a). The forward and
inverse problem thus constitute a mapping from the x axis to the t axis and vice versa,
using the function R(x, t ) as an intermediary.
In order to facilitate a perturbation expansion of the forward problem one needs to
integrate (39) and it is useful to use expression (22) of Corones et a1 (1983):
R ( x , ,t
-
2 x , ) - R(xO,t - 2x0)= - $ E
{,:
ds
j0'-2'
dz V(s)R(s,z)R(s,t
- 2s - T).
(42)
For the forward problem one needs to integrate leftwards; therefore take x,< x0. Setting
t = 2x0, and using (40a) one finds after a rearrangement of variables that:
+
R(x, t ) = - &V(x t / 2 )
+)E
j:" j0
i+Z(.x-s)
ds
dz V(s)R(s,r)R(s,2x - 2s
+t
- 7).
259
The Born approximation in nonlinear inversion
R ( x , t ) can be expressed as a Taylor series in
nth-order scattering effects:
E,
the different powers
E”
indicating the
R(x,t)= ~R,(x,f)
+ &*R2(x,t)
+ ~ ~ R ? ( x +, t ). . . .
(44)
Inserting this expansion in (43) one obtains, equating the coefficients of equal powers
of E ,
RI (x, t ) =
-
4V ( X+ t / 2 ) ,
(450)
R,(x,t ) = 0
1
R,(x, t ) =
(45b)
?jY ds jo
r+r/2
I
2(\
-i)
d t V(s)R,(s,s)Rl(s,2x
- 2s
+t
-
T).
(45c)
The R,(x,t ) for even orders n vanish because only the reflected wavefield is considered.
The reflected waves r(t) for the full medium from x = 0 to x = X have an expansion
similar to (44). Setting x = 0 in (45), and inserting (45a) in (45c) one finds for the
non-zero terms r , and r 3 :
(46a)
r,(t) = - $ V(t/2)
(46b)
In these expressions r l describes the single reflected waves, and r3 describes the waves that
are reflected three times. The generalisations to higher orders is straightforward.
For the inverse problem, one needs to integrate rightward from x = 0. In order to do
so set xo = 0 in (42); using (41) this gives, after a redefinition of variables,
R(x, t ) = r(t + 2x) - + E
it 2(,Y->)
ds
jo
ds V(s)R(s,z)R(s,2x
-
2s
+ t - 5).
(47)
Iterating this equation once, and using (46a) to eliminate the potential, gives for the firstand third-order effects
R ( x , t ) = r(t
+ 2x) + 2E joy
ds jo
r+2(\-~)
dz r(2s)r(z
+ 2s)r(2x + t
(48)
- 7).
The estimated model p(x, E ) = - 4R(x, 0) follows from this expression by setting t = 0.
Inserting the expansions of R(x, t ) and r ( t ) in powers of c in this expression one finds
p(x, E)
jo‘ds jo
2(Y-\)
= - 4Erl(2x) -
4c3r3(2x)- 8~~
dz r , (2s)r,(z
+ 2s)r,(2x
- 7).
(49)
It can be seen from (460) and (46h) that the third-order scattering term r 3 ( t )is being
cancelled by the triple integral of the Born approximation r , (t). The only term contributing to the reconstruction of the potential is the first-order term - 4&r1
(2x). According to
(46a) this gives P(x, E ) = EV(X),as it shouId. The extension of the derivation presented
in this section to higher orders is tedious but straightforward.
5.3. Example 3. The iterative inversion of Morawetz and Kriegsmann (1983)
The inverse problem for the I D PWE (1) has been solved in an iterative way by Morawetz
and Kriegsmann (1983). (A similar iterative inversion scheme for higher-dimensional
problems was presented by Morawetz (1981).) In their formulation the potential is
260
R Snieder
assumed to be an even function, V ( x )= V(-x), which vanishes for 1x1 > xo.The
incident wave also has even parity:
u,(x,t ) = s(t - x) + s(t + x).
(50)
The reflected waves are recorded at x = 0, and the data are given by
(51)
R(t) = u,(0, t ) .
In the formulation of Morawetz and Kriegsmann (1983) a Jost solution P ( x , t ) is used,
which satisfies the PWE (1) with the boundary condition
for x > xo.
P(x, t ) = s(t - ).
(52)
By using a representation theorem which relates the data (at x = 0) via the Jost solution
to the wavefield just after the wavefront (u(x,x)), one can derive the following equation
for the unknown potential (Morawetz and Kriegsmann 1983):
S(t
-
s)(R(r')+ 26'(t))dt
(53)
with
S'(t)=
-
P(0, t ) .
(54)
Applying an integration by parts to (53) and redefining
r(t) =
jo'K(t')dt'
(55)
one finds that
with
d
B ( x , t ) = - [P(x,x - t ) - P( - x, x - t ) ] .
dx
(57)
Equation (56) is a nonlinear integral relation for the potential, because the Jost solution
P ( x , t ) (and hence B ( x , t ) ) is a nonlinear function of the potential, which enters the
right-hand side quadratically. The data r(t) also depend nonlinearly on the potential.
although in practical inversions one considers the data of course as a fixed independent
quantity.
A perturbative analysis can be applied to (56). In order to do so the following
expansions in E can be used, these expansion can be derived from the Neumann series
for the appropriate integral equations
P(0, t ) = P,(O, t )
B ( x , f)
=
+ EP, (0,t ) + E2P,(0,t ) +
B,(x, t ) + E B , ( x ,t ) + t2B2(x,t ) + . . .
r(t>= Erl ( t )
+ E2r2(t)+ . . . .
(58)
The absence of the zeroth-order term for r(t) can readily be inferred from (50),(51) and
( 5 5 ) . The zeroth-order terms Po and Bo are independent of the potential. Inserting these
The Born approximation in nonlinear inversion
c-:L cox
+ c_: jox
26 1
expansions in (56) gives for the first- and second-order terms:
$E~(=
x )E
dzB,(-x, s)P,(O, T
ds
E'
ds
-
s)rl(7)
d? [B,(x,s)P,,(O,z - s)r2(7)
Equating the coefficients from equal powers of E , one sees that the potential depends only
on r i ( t ) ; hence once more it is only the first Born approximation that contributes to the
reconstruction of the potential. The term proportional to E' on the right-hand side is
zero, because there is no corresponding term on the left. This means that the double
reflected waves r2( t ) are being cancelled by repeated iterations of first-order quantities.
This derivation is trivially extended to higher orders. This implies once more that the
nonlinear components in the data do not contribute to the reconstruction of the potential, and that they are cancelled in the inversion by repeated iterations of lower-order
terms.
6. Discussion
Inverse scattering algorithms construct implicitly the potential by subtracting the nonlinear components from the data, and performing a Born inversion of the linear component in the data. Although this point of view does not help us very much in the design
of inversion schemes, it gives us some useful insights in the mechanism of nonlinear
inversion. This conclusion pertains to any nonlinear problem where the forward problem
can be expanded as a regular expansion, and where the inverse problem can be formulated in terms of a sum of operators which act repeatedly on the data (see Snieder
1990). The examples of section 5 illustrate this statement.
The fact that only the Born approximation contributes to the reconstruction of the
potential does not imply that it suffices to perform a Born inversion of the data. The
reason for this is that the data consist of the complete Neumann series instead of the first
Born approximation. Removing the nonlinear components from the data is an essential
ingredient of nonlinear inversion. This removal is achieved in an implicit fashion in any
nonlinear inversion scheme that reconstructs the true potential.
The fact that inverse scattering algorithms subtract the nonlinearities from the data
by iteration of an integral equation implies that there exist interrelations between the
part of the data corresponding to different orders of nonlinearity. (If this were not the
case, then it would be impossible to subtract the nonlinearities by repeated iterations of
the lower-order nonlinearities.) These interrelations are currently poorly understood and
are needed to obtain a better understanding of inverse scattering algorithms.
Errors in the data have a disastrous effect on the performance of inverse scattering
algorithms. For example, it is shown by Koehler and Taner (1977) that timing errors lead
to a severe instability of a Goupillaud inversion for a layered medium. (As shown by
Berryman and Greene (1980), the Goupillaud inversion is equivalent to inverse scattering
using the Marchenko equation (6)) The reason for this is now easy to understand. Errors
in the data destroy the interrelations between the different nonlinear components in the
data, so that the nonlinearities can no longer be subtracted properly. The data are
262
R Snieder
especially sensitive to timing errors, because timing errors destroy the relation between
the arrival times between single reflected waves and multiple reflected waves. The
detrimental effect of data errors on nonlinear inversion is illustrated with a numerical
example in Snieder (1990). In some nonlinear inversions this instability may be partially
countered with some form of regularisation. In an optimisation approach, such as shown
in Snieder and Tarantola (1989) for the imaging of quantum mechanical potentials, such
a regularisation can be incorporated in a natural way.
The stability of inverse scattering algorithms has received considerable interest
(Koehler and Taner 1977, Krueger 1981, Bregman et a1 1985, Carrion 1986). The fact
that the subtraction of nonlinear components in the data leads to instabilities in nonlinear inversion applies to a wide class of nonlinear inverse problems, such as those
presented in section 5. The nonlinear inversion scheme in the examples of Krueger (198 1)
appears to be rather stable. However, the reflection coefficients in the example of
Krueger (1981) are of the order of 5 % , so that one is for practical purposes in the linear
domain. This is confirmed in figure 3 of Krueger (1981) which shows that increasing the
data error by a factor 5 leads to a pointwise fivefold increase in the error of the
reconstructed model, a typical linear phenomenon. In contrast, it is shown in Bregman
et ul (1985) that numerical truncation and round-off effects leads to instabilities if one
is in the strongly nonlinear domain. Similarly, Koehler and Taner (1977) showed that
multiply reflected waves lead to instabilities in the Goupillaud inversion of seismic
reflection data. For a proper appraisal of the stability of a nonlinear inversion scheme,
it is crucial that one is in the nonlinear domain.
Exact inverse scattering methods always rely on a causality principle (Rose et a1 1984,
1985a,b, DeFacio and Rose 1985). Usually this causality principle manifests itself by the
triangularity of some integral kernel. The interpretation of inverse scattering in this
paper elucidates the crucial role of causality in nonlinear inversion. Causality implies
that the linear component of the data arrives before the multiply scattered waves. For
reflection data this is obvious, because a wave that is reflected only once arrives earlier
than a wave that has bounced back and forth in the medium. For transmission data, the
progressing wave expansion shows that the wavefield right after the direct wave depends
linearly on the potential, and that the nonlinear components in the wavefield arrive later
(Burridge 1980, Rose et a1 1984). This means that if one wants to subtract the nonlinearities from the data at time t o , one need only consider the signal at earlier times
t 6 t o . This is implicit in equations (15) where the nonlinearities are subtracted. For
example, in (16d) the quadratic term R, at time r = 2x is being cancelled by a combination of linear terms R, at earlier times t 6 2x.This causality principle also forms the
basis of various kinds of stripping algorithms (e.g. Bojarski 1980, Krueger 1981, Bube
and Burridge 1983, Bregman et a1 1985).
Exact inverse scattering methods were formulated for the Schrodinger equation. In
the Schrodinger equation, the propagation speed of the waves is not affected by the
potential. The formulation of exact inverse scattering methods for media with a variable
wave velocity is problematic. (In one-dimensional inverse problems with a variable
velocity one usually transforms the velocity away by converting depth to travel time (e.g.
Burridge 1980)) For three-dimensional problems with variable wave velocity, an integral
equation for the inverse problem has been formulated by Rose et a1 (1985a) for the
special case where the wave velocity in the inhomogeneity is less than or equal to the
wave velocity in the surrounding reference medium. Cheney et a1 (1989) remove this
limitation using an inverse scattering algorithm which relates the variable velocity
medium to a fast reference medium in such a way that the scattered waves are always
The Born approximation in nonlinear inversion
263
slower than the waves in the reference medium, so that the desired causality properties
can be employed. The results of this paper allow a better understanding of the additional
complications of exact inversion schemes for scattering media with a varying wave
velocity. In inverse scattering, one deals with the nonlinearities by subtracting them from
the data. In a medium with a fixed wave velocity, one can infer the arrival time of a
multiply reflected wave if one knows the arrival times of the single reflected waves. This
is no longer possible if the velocity is unknown because in that case one does not know
at which time one should subtract the nonlinear components from the data. This
problem is aggravated by the fact that for media with a variable wave velocity the
nonlinear components in the data may arrive before the linear components in the data,
which considerably complicates the inversion for media with a variable wave velocity. In
fact, it is not known if the inverse problem for 3D media with an unrestricted variable
wave velocity is well-posed (Cheney et a1 1989).
As noted by Ge (1987), the second iteration of the Marchenko equation produces the
same contribution to the reconstructed potential as the quadratic term of a perturbative
inversion scheme for the Schrodinger equation derived by Moses (1956). It is shown by
Snieder (1990) that the perturbative analysis of the Marchenko equation can be extended
to any nonlinear inverse problem where both the forward and inverse problem can be
expressed as regular Taylor series. It is therefore not surprising that the iterative solution
of the Marchenko equation leads to the same algorithm as for a perturbative inversion
as developed by Moses (1956). It follows from the fact that the arguments presented in
this paper can be extended to a very wide class of nonlinear inverse problems (Snieder,
1990) that the subtraction of the nonlinearities and the reconstruction of the
inhomogeneity by the linear component in the data is a fundamental aspect of nonlinear
inversion.
Acknowledgment
The constructive remarks of the reviewers are greatly appreciated. This research was
partly supported by the European Communities (DG XII).
Appendix: the elimination of the quadratic nonlinearities in the Newton-Marchenko
method
The second-order contribution to the solution K of the Newton-Marchenko equation
(27) is by analogy with (156) given by
d2n’R,(a, 2 , ii‘, v )
+
I:’ i
dp
d2n‘Rl( x
+ B, ii, ii‘, v)Kl(B, ii’, v ) .
(All
Inserting the second-order Born approximation (30b) in (26) gives
x exp[ikii’ * (v
-
x)] exp[ikii * (r’ - x)] exp(iklr - v ‘ l ) exp( - ikx).
(A2)
R Snieder
264
Carrying out the integration l d 2 n ’ over the unit sphere gives
-
x sin(k1r’ - xi) exp[ikii (r - x)] exp(ik1v
1) exp( - ika).
(A31
The second term on the right-hand side of ( A l ) follows by using (31) for K , , and by
inserting the first Born approximation (30a) in (26). This gives
jnxdfl 1d2n’R,
(a
+ P, ii, ii’, x ) K ,(P, A’,x) = 4
jOz
d b jd2n‘ j
-:
64.n
x exp[ik, ii’
x
- I*’
- (r - x)] exp[ik, ii
*
(U -
x)] exp[ - ik, ( a
dk, k,
+ /3)]
j
d3r V ( r )
J
dk,
-%
j
d3rfV(v’)
sin ( k , Iv’ - XI)
-exp[ik,(ii’
Ir’ - XI
(v’ .- x)
-
b)].
(‘44)
Expanding the sine in two exponentials, and integrating over k, gives two delta
functions, S(P - in’ (r’ - x) - lr’ - XI) and 6(8 - n’ (r’ - x) jr’ - XI ) . The last delta
function does not contribute since f i > 0 and ii’ (v’ - x) - lv’ - XI < 0. This gives
-
x exp[ik, it’
(v
-
x)] exp[ik, ii ( r
-
+
x)] exp[ - ik, (SI
+ P)]
Carrying out the integration over fl and over the unit sphere Sd’n’ produces
joJ-d p
d2n’R,( 3 + 8, ii, ii’, x ) K ,(P, ii’, x)
x sin(klr - r’ 1) exp[ikii
=
4
{ dk 1d3r
16.n - =
x)] exp( - iklv’
d3r’
V(v)V(r’)
lr
-
v’/Iv’ - x /
xi) exp( - ika).
(A61
Adding (A3) and (A6), and setting a = 0, one finds for the total contribution of the
second-order contributions:
(U -
x [sin(klr’ - xi) exp(ik1v - r’ 1)
-
+ sin(k1r - r’ 1)
exp( - ikiv’ - XI ) ] .
(A7)
Expanding the sines in exponentials one obtains four terms, two of which cancel each
other. The remaining terms can be recombined to give
x exp[ik(lr
v’ I
+
- xi)]
{exp[ikii * ( r - x)]
[-ii]}
(‘48)
where [ - i t ] stands for the same term with ii replaced by - fi. Defining r , E v - v‘ and
r‘ - x, and carrying out the k integration, one finds
-
11”’
18,
K,(a
= O,ii,x) =
~
1
V(v,
jd’r, (d3r2
1671
+ v, + x)V(vz + x)
rl r2
-
The Born approximation in nonlinear inversion
265
Carrying out the rI and r2 integrations in polar coordinates, gives with the definitions
uI = (ii * ? I ) and u2 = (it * r 2 ) :
X
V(rl
+ r2 + x)V(r2 + x)
TI
x [6(rl
r?
+ r2 + u l r l + u 2 r 2 )
-
6(r,
+ r2
-
u l r I - uzr,>].
(A 10)
The delta functions give a contribution equal to zero upon integration over the two
variables ul and u2. This means that
& ( a = 0,it, x) = 0.
(A1 1 )
This confirms that in the Newton-Marchenko algorithm the second-order Born
approximation is being cancelled by a repeated iteration of the first-order Born
approximation.
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~